Preheat oven to 350 degrees. Line a 9x5 inch bread pan with nonstick foil and spray with nonstick spray.
In a medium bowl whisk together flour, cinnamon, cloves, ginger, salt, and baking soda. In a large bowl cream together butter, sugar, and applesauce.
Add dry ingredients to wet ingredients and mix until blended. Pour mixture into prepared baking pan. Bake for 45-55 minutes until a toothpick inserted into the center comes out clean. Allow to cool on a cooling rack.
For the frosting, in a medium bowl cream together butter, cream cheese, powdered sugar, vanilla, and milk until smooth and spreadable. Spread over cooled gingerbread loaf. Allow to set up for 1 hour (or if you can't wait you can serve it immediately, but I like it best when the frosting has a chance to set up a bit!). Store in airtight container.